ESTIMATION OF EFFICIENT REGRESSION MODELS
FOR APPLIED AGRICULTURAL ECONOMICS RESEARCH
by
Octavio A. Ramírez
Sukant K. Misra
and
1
Jeannie Nelson
1
Associate Professors and M.S. student, respectively. Department of Agricultural and Applied
Economics, Texas Tech University, Box 42132, Lubbock TX 79409-2132. Phone: (806) 742-2821,
Fax (806) 742-1099, Email (corresponding author): Octavio.Ramirez@ttu.edu
2002 Annual Meeting of the American Agricultural Economics Association
Long Beach, CA, July 28-31, 2002
ABSTRACT
This paper proposes and explores the use of a partially adaptive estimation technique to improve the reliability
of the inferences made from multiple regression models when the dependent variable is not normally
distributed. The relevance of this technique for agricultural economics research is evaluated through Monte
Carlo simulation and two mainstream applications: A time-series analysis of agricultural commodity prices
and an empirical model of the West Texas cotton basis. It is concluded that, given non-normality, this
technique can substantially reduce the magnitude of the standard errors of the slope parameter estimators in
relation to OLS, GLS and other least squares based estimation procedures, in practice, allowing for more
precise inferences about the existence, sign and magnitude of the effects of the independent variables on the
dependent variable of interest. In addition, the technique produces confidence intervals for the dependent
variable forecasts that are more efficient and consistent with the observed data.
Key Words: Efficient regression models, partially adaptive estimation, non-normality, skewness,
heteroskedasticity, autocorrelation.
Multiple regression models are used in applied agricultural economics research with two main
purposes: forecasting and making statistical inferences about the effect of exogenous variables on the
dependent variable. Efficient estimation of the model coefficients is important in both cases. Slope
parameter estimators with lower standard errors represent more precise measurements of the magnitude of
the impacts of the exogenous variables on the dependent variable, and produce more reliable predictions.
Ordinary least squares (OLS) is widely used in empirical work because if the model’s error term is
normally, independently and identically distributed (n.i.i.d.), OLS yields the most efficient unbiased
estimators for the model’s coefficients, i.e. no other technique can produce unbiased slope parameter
estimators with lower standard errors. Maximum likelihood (ML) based on the n.i.i.d. assumption is
equivalent to OLS. Generalized least squares (GLS) can be used to improve estimating efficiency relative
to OLS when the error term is heteroskedastic, autocorrelated or both. Even more efficient slope parameter
estimators can be obtained in this case through ML, by specifying a likelihood function in which the error
term is assumed normal, but not i.i.d. (Judge et al.). This is commonly known as “correcting” for
heteroskedasticity or autocorrelation.
If the dependent variable and, thus, the error term is a continuous but not normally distributed
variable, however, OLS (in the i.i.d.-error case) or normal-error ML (in the non-i.i.d.-error case) is not the
most efficient way of estimating the slope parameters of a multiple regression model (Judge et al.). Since
non-normal dependent variables are not uncommon in applied modeling work, Goldfeld and Quandt argue
vehemently against the continued reliance on the assumption of error term normality for estimating
regressions. Three approaches are currently available for estimating multiple regression models under nonnormality: robust, partially adaptive, and adaptive estimation (McDonald and White).
Robust estimators are based on minimizing some function of a scale-adjusted error term that gives
less weight to large error values. They can be asymptotically more efficient than OLS when the tails of the
underlying error term distribution are “thicker” than the normal (McDonald and Newey). Least Absolute
Deviation (LAD) is an example of a robust estimator that is asymptotically more efficient than OLS for
many thick-tailed error-term distributions. Other robust estimators include the M and the L-estimators
(Bickel, 1976; Mosteller and Tukey, 1977; Bierens, 1981; and Dielmam and Pfaffenberger, 1982; Joiner
and Hall, 1983). Fully adaptive estimators involve non-parametric estimates of the derivative of the log of
the unknown density. They have the advantage of being asymptotically efficient, as the maximum likelihood
estimator obtained using knowledge of the actual error term distribution. Given that the correct error term
distribution is seldom known, fully adaptive estimation should be preferred when working with large
samples, since it would produce the lowest possible standard errors. Examples of fully adaptive estimators
include Hsieh and Manski’s Adaptive Maximum Likelihood (AML) estimator, which is based on a normal
kernel density; and Newey’s generalized method of moments estimator. For more details about these
techniques please see McDonald and White. The more favorable asymptotic properties of fully adaptive
estimators, however, are meaningless when working with small samples.
Partially adaptive estimators are ML estimators based on specific families of error term
distributions, in hopes that the assumed family is flexible enough to accommodate the shape of the true
unknown distribution of the error. Partially adaptive estimators based on the t distribution (Prucha and
Kelejian), the generalized t (McDonald and Newey), and the generalized beta distribution (McDonald and
White) have been explored in the econometrics literature. Partially adaptive estimators are asymptotically
efficient only if the true error term distribution belongs to the family of the assumed distribution.
However, McDonald and White show that partially adaptive estimators based on assuming a
flexible family of densities that can accommodate a wide variety of distributional shapes can substantially
outperform OLS and all available robust and fully adaptive estimators in small sample applications
involving non-normal errors.
In other words, when estimating a multiple regression model with a
continuous but non-normally distributed dependent variable and a small sample size, a partially adaptive
estimator based on a flexible distribution that can reasonably approximate the true underlying error term
distribution would likely produce slope parameter estimators with the lowest possible standard errors,
potentially much lower than OLS.
Intuitively, and improved modeling of the error term distribution, i.e. of the distribution of the
deviations from the observations from the multiple regression hyper plane, allows for a higher degree of
certainty about the location of the hyper plane, i.e. about the parameters determining that location. If, for
example, the error term distribution is substantially right skewed, OLS (which implicitly assumes a
normally distributed error) cannot account for the extreme positive deviations that characterize rightskewness. As a result, the location of the regression hyper plane would be less certain that when assuming
an error term distribution that can account for these deviations.
The use of partially adaptive procedures for increasing slope parameter estimation efficiency
through a more precise modeling of the error term distribution, however, has not been explored in the
agricultural economics literature. Given the importance of obtaining more precise estimates of a model
parameters for forecasting and statistical inferences, partially adaptive estimation techniques could be very
useful in applied agricultural economics research. Applied researchers are increasingly better trained in
basic econometric techniques, including the standard maximum likelihood estimation procedures required to
implement partially adaptive estimation. This should facilitate the adoption of more efficient modeling
techniques by applied researchers.
However, a practical shortcoming of the available partially adaptive estimators is that they were
not designed to model heteroskedasticity or autocorrelation. Since, in addition to inefficient slope parameter
estimators, unchecked heteroskedasticity or autocorrelation leads to biased and inconsistent standard error
estimators, this limits the applicability of partially adaptive estimation when the error term is non-normally
distributed and non-i.i.d. In other words, an applied researcher concerned about efficiency would have to
choose between partially adaptive estimation or “correcting” for the non-i.i.d. error. In addition, available
partially adaptive estimators cannot be straightforwardly used in multiple-equation (i.e. seemingly unrelated
equation –SUR–) set up, which is a common procedure to increase estimation efficiency in relation to OLS.
This paper addresses the former issues by proposing and evaluating the theoretical and empirical
performance of a partially adaptive estimator that can jointly model error term non-normality,
heteroskedasticity and/or autocorrelation in single and multiple equation specifications. The technique is
based on assuming that an expanded form of the Johnson SU family of distributions (Johnson, Kotz, and
Balakrishnan) can approximate the true underlying error term distribution. The Johnson SU family has been
previously applied by Ramirez to simulate non-normally distributed yield and price distributions for
agricultural risk analysis. Through Monte Carlo simulation assuming a variety of scenarios, it is shown
that when the underlying error term is non-normally distributed and non-i.i.d., the proposed estimator can
substantially increase slope parameter estimation efficiency in comparison to OLS, GLS (normal-error
ML), and all other partially adaptive estimators available in the econometrics literature. The proposed
technique is also validated and illustrated through two agricultural time series modeling applications.
The Estimator
The proposed partially adaptive estimator is obtained by assuming that the model’s error term (U)
follows the following expanded form of the Johnson SU family of distributions:
(1)
Y = Xβ + U,
(2)
U = σ{sinh(ΘV)−F(Θ,µ)}/{ΘG(Θ,µ)}, V ∼ N(µ,1),
2
F(Θ,µ) = E[sinh(ΘV)] = exp(Θ /2)sinh(Θµ), and
2
2
2 1/2
G(Θ,µ) = [{exp(Θ )−1}{exp(Θ )cosh(−2Θµ)+1}/2Θ ] ;
where Y is an n×1 vector of observations on the dependent variable; X is an n×k matrix of observations on
k independent variables including an intercept; β is a k×1 vector of intercept and slope coefficients;
−∞<Θ<∞, −∞<µ<∞, and σ>0 are transformation parameters; and sinh(x) and cosh(x) are the hyperbolic
sine and cosine functions, respectively. Using the results of Johnson, Kotz and Balakrishnan (pp. 34-38) it
can be shown that in the model defined above:
(3)
2
E[U] = 0, Var[U] = σ ,
3
½
2
3/2
Skew[U] = E[U ] = S(Θ,µ) = −1/4w (w−1) [w{w+2}sinh(3Ω)+3sinh(Ω)]/G(Θ,µ) ,
Kurt[U]
=
4
E[U ]
=
K(Θ,µ)
=
2
2
4
3
2
2
{1/8{w−1} [w {w +2w +3w −3}cosh(4Ω)+4w {w+2}
2
cosh(2Ω)+3{2w+1}]/G(Θ,µ) }−3;
2
where w = exp(Θ ) and Ω = −Θµ. The equations in (3) indicate that, in this model specification, Xβ solely
2
determines E[Y], σ independently controls Var[U], and µ and Θ determine error term skewness and
2
kurtosis. Thus, standard heteroskedastic specifications can be introduced by making σ a function of the
variables influencing Var[U], without affecting the E[Y] or the error term skewness or kurtosis. Evaluation
of Skew[U] and Kurt[U] shows that if Θ≠0 but µ=0 the distribution of U is kurtotic (i.e. “fat-tailed”) but
symmetric. The sign of Θ is irrelevant, but higher values of Θ cause increased kurtosis. If Θ≠0 and µ>0, U
has a kurtotic and right-skewed distribution, while µ<0 results in a kurtotic but left skewed distribution.
Higher values of µ increase both skewness and kurtosis, but kurtosis can be scaled back by reducing |Θ|.
In short, a wide variety of right and left skewness-kurtosis coefficient combinations can be
obtained by altering the values of these two parameters. Also, if µ=0, S(Θ,µ)=0, and the former becomes
symmetric but kurtotic error term model. Further, as Θ goes to zero, U approaches σV, Var[U] approaches
2
σ and K(Θ,0) also becomes zero, indicating that the normal-error regression model is nested to this nonnormal error model. As a result, in applied regression analysis, if the error term is normally distributed,
both µ and Θ would approach zero and the proposed estimator for the slope parameter vector β would be
the same as OLS. Also, the null hypothesis of error term normality (i.e. OLS) vs. the alternative of nonnormality can be tested as Ho: Θ=0 and µ=0 vs. Ha: Θ≠0 and µ≠0. The null hypothesis of symmetric nonnormality versus the alternative of asymmetric non-normality is Ho: Θ≠0 and µ=0 vs. Ha: Θ≠0 and µ≠0.
To specify a non-normally distributed and autocorrelated error term model, consider a model with
2
an n×1 error term vector U, which is normally distributed but not i.i.d. Following Judge, et al., let Φ = σ ψ
-1
be the covariance matrix of the error term vector, P be an n×n matrix such as P’P = ψ , Y* = PY (an n×1
vector), and X* = PX (an n×k matrix), where Y and X are the vector and matrix of original dependent and
independent variables. Given the choice of P, the transformed error term U* = PU = P(Y−Xβ) =
(PY−PXβ) = (Y*−X*B) is i.i.d. Under the assumption of normality, the log-likelihood function that has to
be maximized in order to estimate a multiple regression model with non-i.i.d. errors then is:
(4)
2
2
NLLj = −(n/2)ln(σ ) −0.5×ln |ψ | −(U*’U*/2σ ).
A regression model that can accommodate non-normality (kurtosis and right or left skewness), and
autocorrelation and/or heteroskedasticity, is finally obtained by applying the transformation in equation (2)
to U*. When the error term vector U = (Y−Xβ), is autocorrelated, a first transformation is used to obtain a
non-autocorrelated error term U* = PU = (PY−PXβ) = (Y*−X*β). This is then transformed to a normal
error term vector V through equation (2). If U is only heteroskedastic, P is an n×n identity matrix. The
concentrated log-likelihood function to be maximized in order to estimate this model’s parameters is:
(5)
n
n
2
NNLL = −0.5×ln |ψ | + Σ ln(Gi) −Σ 0.5×Hi }; where:
i=1
i=1
2 -1/2
Gi = G(Θ,µ)(1+Ri ) /σ;
–1
Hi = {sinh (Ri)/Θ}−µ;
–1
Ri = {G(Θ,µ)ΘUi*/σ}+F(Θ,µ);
2 1/2
and i=1,…,n refers to the observations, sinh (x) = ln{x+(1+x ) } is the inverse hyperbolic sine function,
and F(Θ,µ) is as given in equation (2). The first and second terms in equation (5) are the natural logs of the
Jacobians of the first and second transformations, respectively. Hi is the inverse of the transformation to
normality in equation (2). If the error term is believed to be autocorrelated, P and |ψ | must be specified to
make equation (5) operational. Judge, et al. derives P and |ψ | for first- and higher-order autoregressive
2
2
processes. As before, in this autocorrelated specification, E[Y]= Xβ and Var[U]= σ ψ . Thus, σ can still
be used to model systematic changes in the variance of U across the observations without affecting
skewness or kurtosis.
The multiple equation (SUR) equivalent of equation (5) is obtained by applying a set of normality
transformations [equation (2)] to a set of m “transformed” n×1 non-normal random errors, Uj* =
Yj*−Xj*βj (j=1,...,m), where Yj*=PjYj, Xj*=PjXj, and Yj and Xj are the original vectors and matrices of
dependent and independent variables, respectively. As in the single equation case, the transformed set of
random vectors Vj follow a multivariate normal distribution with means µj (j=1,...,m) and covariance
matrix Σ. The non-diagonal elements of Σ (ρjk) account for the correlation between the error terms of the m
equations. The concentrated log-likelihood function for this model is a straightforward multivariate
extension of equation (5):
(6)
m
n m
n m
-1
MNNLL = { −(n/2)×ln|Σ| −0.5×∑[ln(|ψ j|)] +∑ ∑ [ln(Gji)] −0.5×∑ ∑ [{Hi*(Σ )}.*Hi]};
j=1
i=1 j=1
i=1 j=1
where Σ is an m×m matrix with unit diagonal elements and non-diagonal elements ρjk; Gji is as defined in
-1
equation (5) if Yj (and thus Yj*) is not normally distributed or Gji=σj if Yj is normally distributed; and Hi
is a 1xm row vector with elements Hji (j=1,...,m) also defined in equation (5) if Yj is not normally
distributed and Hji=(Yji*–Xji*βj)/σj if Yj is normally distributed. The operator * indicates a matrix
multiplication; and .* indicates an element-by-element matrix multiplication.
The multivariate log-likelihood function {equation (6)} simply links m univariate functions
[equation (5)] through the cross-error term correlation matrix Σ. As in the normal error case, if some of the
m dependent variables of interest are correlated to each other, using equation (6) to jointly estimate the βj
vectors should result in an improved efficiency in comparison with the case where they are estimated
separately. Maximum likelihood estimation is conducted by finding the values of the parameters (Θj, µ j, βj,
and those in the Σ, Pj, and ψ j matrices) that maximize the log-likelihood function [equation (6)]. This is
achieved through numerical optimization procedures, such as the Newton-Raphson algorithm, which are
available in most econometric software packages, including Gauss 386i. These pre-programmed procedures
only require a few standard command lines and the log-likelihood function. In addition to parameter estimates,
they provide standard errors based on a numerical estimate of the Hessian matrix of this function.
Monte Carlo Simulation Analysis
The sample design used by Hsieh and Manski, Newey, and McDonald and White was adopted for
the Monte Carlo simulation to ensure comparability with previous results. For the first phase of the
simulation, the regression model is given by:
(7)
Yji = βj0 + βj1Xji + Uji = -1 + Xji + Uji (j=1);
where the explanatory variable Xji = 1 with a probability of 0.5 and Xji = 0 with a probability of 0.5. Xji is
also assumed to be statistically independent of Uji. Thus, each model can be interpreted as estimating a shift
parameter that separates two identical distributions except for a location parameter. The specifications for
the error term distribution are taken from McDonald and White: (a) Normal {N(0,1)}; (b) Mixture of
normals or variance-contaminated normal {0.9*N(0,1/9) +0.1*N(0,9)}; and (c) Lognormal. The
distributions were re-scaled and shifted, when necessary, to be drawn from a parent population with unitary
variance and zero mean. One thousand samples of size 50 were generated for each simulation experiment,
using the same X’s for each sample. For the second phase of the simulation, first order autocorrelation was
induced by multiplying each of the simulated error terms by the inverse of the P matrix defined above, for
two different ρ values of 0.5 and 0.8. Heteroskedasticity was induced by multiplying the errors by I+0.5,
where I is a binomial index variable taking values of zero or one with equal probability. For modeling
purposes I is assumed to be a known qualitative factor that shifts the variance of the dependent variable
from 0.5 to 1.5. In both cases the error term distributions maintain a zero mean and unit variance.
Simulation Results and Estimator Performance
As in previous studies (McDonald and White; Newey; Hsieh and Manski) the root mean squared
error (RMSE) of the slope estimators is used as the criteria for comparing the relative performance of
different estimators. The results of McDonald and White for three underlying error term distributions:
normal, kutotic-only and kurtotic and skewed (Table 1), are used as a basis for comparison. Under OLS,
the RMSE of the slope estimators is always around 0.28. When the true error term is i.i.d. normal, the
proposed estimator, yields maximum likelihood parameter estimates for Θ and µ that are zero or not
statistically different from zero, and slope estimates that are identical to those from OLS. Most of the
estimators explored in McDonald and White perform similarly or slightly worse than OLS, as expected,
since OLS is the most efficient estimator under i.i.d. normal error-term conditions.
The variance-contaminated normal implies a symmetric unimodal but thick-tailed error-term
distribution with a kurtosis coefficient of approximately 20 in this case. The proposed estimator performs
relatively well in estimating the slope coefficient (β2), with a RMSE of 0.115 (Table 2). Nine other
estimating techniques provide comparable efficiency levels, but none produces a RMSE lower than 0.11
(Table 1).
A log-normal error term distribution is unimodal but not symmetric. It exhibits positive degrees of
kurtosis and skewness. The RMSE of the proposed slope-parameter estimator is 0.050, approximately 20%
of the RMSE of the OLS estimator. According to McDonald and White criteria, this is an “impressive”
performance, only comparable to the EGB2’s 0.05 RMSE. All other alternative estimation techniques
produce RMSE’s of 0.11 and above.
In summary, the estimator proposed in this study performs very favorably in comparison with the
16 estimating techniques evaluated by McDonald and White according to the standards applied by these
authors and independently established in previous studies (Newey; Hsieh and Manski). It clearly excels
when the underlying error-term distribution is asymmetric. This implies a potential for efficiency gains
when
using
the
proposed
estimator
under
conditions
of
error-term
non-normality
and
heteroskedasticity/autocorrelation, or in a disturbance related equations set up.
When the variance-contaminated error-terms are simulated to be heteroskedastic, the slope
estimation efficiency (i.e. the RMSEs) of the EGB2 and of the proposed estimator are not substantially
affected, and the modeling of heteroskedasticity with the proposed estimator, by letting σ =α1+α2I (where I
is the binomial index variable known to affect the error-term variance), only produces modest efficiency
gains (Table 2). Under log-normal heteroskedastic errors, however, the EGB2 and the i.i.d form of the
proposed estimator are substantially less efficient in comparison to the log-normal homoscedastic scenario.
Their RMSEs increase from 0.054 and 0.050 to 0.117 and 0.104, respectively. In contrast, a RMSE of
0.202 is obtained when estimating a heteroskedastic but normally distributed error-term model specification
by maximum likelihood. The modeling of heteroskedasticicy with the non-i.i.d. form of the proposed
estimator recovers the lost estimation efficiency gains. The resulting RMSE of 0.052 is 25% of the RMSE
obtained with the heteroskedastic normal model.
In contrast, failure to model autocorrelation (ρ=0.5 and ρ=0.8) reduces the slope estimation
efficiency of all estimators. The RMSEs increase to 0.320 and 0.388, respectively, when OLS is used
under a variance-contaminated autocorretaled error-term, and to 0.334 and 0.439 under a log-normal
autocorrelated error-term. The i.i.d. form of the proposed estimator limits those RMSE increases to 0.141
and 0.188 (variance-contaminated autocorrelated), and to 0.126 and 0.260 (log-normal autocorrelated).
Comparable RMSEs are found under the EGB2.
In short, the efficiency gains in the estimation of the slope parameter resulting from the use of
either the EGB2 or the i.i.d. from of the proposed estimator, in relation to OLS, are reduced to about 50%
if the underlying error-term is non-normally distributed and autocorrelated. The relative reduction is even
more severe in comparison to the standard maximum likelihood autocorrelated normal-error estimator and,
in the 0.8 autocorrelation log-normal error term scenario, the later is actually more efficient (Table 2). This
illustrates the importance of a partially adaptive estimator that can account for both error term nonnormality and autocorrelation/ heteroskedasticity: The autocorrelated form of the proposed estimator fully
recovers the efficiency losses, producing slope-estimator RMSEs that are similar to those obtained under
the i.i.d. non-normal underlying error-term scenarios. These RMSEs are 50 to 75% lower than those
obtained with the standard maximum likelihood autocorrelated normal estimator and 25 to 80% lower than
those obtained under the EGB2 or the i.i.d. form of the proposed estimator (Table 2).
Agricultural Economics Applications
Simple Time Series Models of Agricultural Commodity Prices
An issue of interest for agricultural economists is whether real commodity prices have been declining
through time and if price variability has changed over time making the production of a particular crop more risky.
Both, the normal-error regression model and the proposed partially adaptive (i.e. non-normal error) regression
model are used to analyze this issue in the case of annual (1950-1999) U.S. corn and soybean prices.
The two price series are adjusted for inflation to the year 2000 using the producer price index for all
agricultural products (USDA/NASS, http://www.usda.gov /nass/, March 2001). Both series are stationary
according to augmented Dickey-Fuller unit root tests. OLS models assuming second-degree polynomial time
2
trends (i.e. Xβ = β0 + β1 t + β2t ; t=1,…,50) are first estimated using Gauss 386i lreg procedure (Table 3).
Durbin-Watson tests indicate first order positive autocorrelation in both cases. Therefore, standard tenth-order
autoregressive models are estimated using Gauss 386i autoreg procedure. Since the autocorrelation parameter
estimates decrease in size and the second and higher-order autocorrelation coefficients are not statistically
different from zero at the 10% level, it is concluded that a first order autoregressive [AR(1)] error term
specification is sufficient to correct for autocorrelation.
The estimated AR(1) models are presented in Table 3. Note that, although the autocorrelation
coefficients are statistically significant at the 1% level in both the corn and soybean price models, the standard
error estimates for the estimators of the slope parameters βc1, βc2, βs1, and βs2 are larger under the AR(1) models.
This is a common occurrence in applied modeling work, not withstanding of the fact that, under nonindependently distributed errors, the AR(1) slope parameter estimators are theoretically more efficient than OLS.
The cause of this apparent contradiction is that, in the presence of autocorrelation, the OLS standard error
estimators are biased, generally downwards, i.e. the OLS standard error estimates tend to underestimate the
correct standard errors and, thus, are not reliable.
Theoretically, if corn and soybean prices are correlated through time, more efficient slope parameter
estimators can be obtained by estimating the two models jointly, using a seemingly unrelated regression (SUR)
procedure. This procedure is available in Gauss, SAS, and other statistical analysis software under the
assumption of independently distributed errors, but not under autocorrelated errors. However, the log-likelihood
function used to estimate such a model (available from the authors upon request) is a straightforward bivariate
extension of equation (4). In the case of the soybean price model, the estimated standard errors for the estimators
of the regression slope parameters are moderately lower under this SUR-AR(1) procedure, but they are slightly
higher in the corn price model. The overall efficiency gains from SUR estimation are likely limited by the low
corn-soybean price correlation coefficient estimate under the assumption of error term normality. The SURAR(1) model is then expanded to evaluate if the variances of the price distributions have been changing through
time. The two most common time-dependent heteroskedastic specifications are evaluated:
(8)
2
σj = σj0 + σj1t, and σj = σj0 + σj1t, t = 1,…,50, j = c and s;
which make the standard deviation or the variance of the corn and soybean price distributions linear
functions of time, respectively. Likelihood ratio and t tests do not reject Ho: σj1 = 0 at the 20% level of
statistical significance, suggesting that the error terms are homoskedastic in both cases. Since βc1 and βc2 are not
statistically significant at the 20% level, and second degree polynomial regression specifications are
susceptible to multicollinearity, a final model [NSUR-AR(1)] is estimated without βc2 (Table 3), in which
the linear time-trend parameter βc1 is statistically different from zero at the 5% level.
The NSUR-AR(1) model is the most statistically efficient model currently achievable by an applied
researcher dealing with the simple issue of whether real commodity prices have been declining over time.
However, the previously discussed Monte Carlo simulation results suggest that the non-normal regression
modeling techniques described above offer the potential for increased efficiency, i.e. slope parameter
estimators with lower standard errors. The NSUR-AR(1) model is expanded to allow for the possibility of
error term skewness and kurtosis. This expanded model [NNSUR-AR(1)] is estimated by maximizing the
concentrated log-likelihood function given in equation (6), setting m=2. This log-likelihood is a function of
the same parameters as the NSUR-AR(1) model, plus the parameters accounting a potential non-normality in
the error term distributions of the corn (Θc and µ c) and soybean (Θs and µ s) models.
The statistical significance of the non-normality parameters is verified through the most reliable
likelihood ratio tests. The concentrated log-likelihood function for the NSUR-AR(1) model reaches a
maximum value of 36.92, versus 52.55 in the case of the NNSUR-AR(1) model. Thus, the likelihood ratio
2
test statistic for Ho: Θc=µ c=Θs=µ s=0 is χ (4) = -2×(36.92-52.55) = 31.26 allows for rejection of Ho at the
1% level. The non-normal SUR-AR(1) regression model is statistically superior to the normal SUR-AR(1)
model. If the NNSUR-AR(1) model is restricted to only allow for non-normality in the error term of the corn
2
price regression, the maximum value of the log-likelihood function decreases from 52.55 to 42.89, resulting a χ (2)
2
test statistic of -2×(42.89-52.55) = 19.32, and rejection of Ho: Θs=µ s=0 at the 1% level. Alternatively, the χ (2)
test statistic for Ho: Θc=µ c=0 is -2×(46.12-52.55) = 12.86 rejects Ho at the 1% level as well. The previous
results indicate that the error terms of both the corn and the soybean price models are non-normally distributed.
The powerful D’Agostino-Pearson omnibus (K2) normality test can be applied to the OLS
residuals to confirm these results. This test is based on a standardized combination of the well-known
skewness and kurtosis coefficients, which, on the one hand, allows for rejection of the null hypothesis of
normality on either count (skewness or kurtosis) and, on the other, it avoids the “double jeopardy” of
separate skewness and kurtosis tests. The K2 statistic is 11.93 for the corn price residuals and 8.26 for the
2
soybean price residuals. Since under the null hypothesis of normality K2 is distributed as a χ (2), normality is
rejected at the 1% significance level in both cases. Non-normal error term models are strongly justified.
However, the estimated value for the skewness parameter in the corn model (µ c) is very low (-0.0539) and
2
statistically insignificant at the 20% level. A likelihood ratio test (χ (1) = -2×(52.54-52.55) = 0.02) confirms
this result. Therefore, µ c is set equal to zero in the final NNSUR-AR(1) model (Table 3).
As explained earlier, µ c=0 implies that the distribution of the error term in the corn model is
kurtotic but not skewed, while the positive sign of the parameter estimate for µ s indicates positive, i.e. right
skewness in the error term distribution of the soybean model. The specific skewness and kurtosis coefficients,
calculated using the formulas in equation (3), are 0 and 22.20 (corn), and 2.08 and 8.61 (soybeans), respectively.
Right skewness, i.e. upward price spikes that are relatively more pronounced than lower price occurrences, is not
a surprising finding, and can be visually perceived on a scatter plot of the soybean price data (Figure 1). The
simulation results are empirically validated in this case: The estimated standard errors for the slope parameter
estimators, are substantially lower under the non-normal model (0.39, 2.81, and 5.30 vs. 0.56, 3.83 and 7.50
under the normal), which means that the proposed partially adaptive estimator allows for much narrower (i.e.
more precise) confidence intervals for the rates of decreases of corn and soybean prices through time.
Confidence intervals for the price occurrences involve the uncertainty in the estimation of all of the
model parameters as well as the estimated error term distribution. These are obtained by applying the
numerical technique of Krinsky and Rob. This technique is based on the asymptotic properties of the ML
estimators, i.e. on the fact that they are normally distributed, consistent, and with known covariance matrix.
It uses the same principle applied to construct confidence intervals for the true parameter values to
numerically simulate the probability distribution from which they could have been drawn. Specifically, let
^
Γ i be the kx1 vector of maximum likelihood estimators for Γi, the vector of true population parameters
^
^
underlying the normal (i=N, k=10) or the non-normal (i=NN, k=14) regression model, and CM [Γi ] be the
^
^
estimated covariance matrix for Γ i . Then, the joint probability distribution of Γ i is simulated by:
^
(9)
^
~
Si = Z x Chol( CM [Γi ] ) + Γ i ,
where Z is an mxk matrix of independently distributed standard normal random variables, Chol(.) denotes the
~
^
Cholesky decomposition, and Γ i the kx1 vector of parameter estimates obtained from Γ i . Equation (9) yields
^
~
^
an mxk matrix of random variables with mean Γ i and covariance matrix CM [Γi ] . Since, under a correct
^
^
^
^
model specification, Γ i is a consistent estimator for Γi and CM [Γi ] is a consistent estimator for CM [Γi ] , Si
is a theoretically correct probabilistic statement about Γi. Thus, the boundaries of a (1-α)% confidence
interval for the expected price under the normal (non-normal) model at time period t can be obtained by
extracting the m sets of simulated parameter values from SN (SNN), using them to obtain m “predicted” price
values for time t, and finding the (α/2) x m and the [(1-α)+α/2] x m largest of these m price values.
th
th
Confidence intervals for the actual price realizations require simulation of m error term draws as well.
In the case of the normal regression model, these are obtained by extracting the m simulated values for the
standard deviation parameter (σc or σs) from SN and multiplying them by m independent draws from a
standard normal random variable. In the case of the non-normal model, the m sets of simulated values for σc,
µ c and Θc (or σs, µ s and Θs) have to be extracted from SNN and coupled with m independent standard normal
draws. Then, m non-normal error term values are simulated by applying equation (2). The final step in
constructing the boundaries of a (1-α)% confidence interval for the actual price observations is to add the m
simulated error term values to the corresponding m price “predictions” and find the (α/2) x m and the [(1th
th
α)+α/2] x m largest of the resulting m simulated price realization values. The former process is programmed
in Gauss 386i with m=100,000 and repeated for t=0,…49 under the normal and non-normal corn and soybean
price models. The resulting boundaries are joined to obtain (1-α)% confidence bands for the 50 individual
price predictions and for the 50 individual price realizations observed from 1951 to 2000.
The 95% confidence bands for the soybean price predictions, i.e. for the expected soybean prices, and
for the actual price occurrences under the normal and non-normal regression models are presented in Figures 1
and 2. The predictions, and the corresponding confidence bands, do not follow the typical curvilinear shape of
a second-degree polynomial in this case because they are autocorrelated predictions (Judge, et al., p. 316). The
95% bands for the 50 soybean price predictions under the normal model show an average width of $0.714/bu
versus $0.557/bu under the non-normal model, i.e. they are 28% wider, on average. The 95% confidence
bands for the 50 corn price predictions are 35% wider under the normal model than under the non-normal
model, on average. An evaluation of the 80% to 95% confidence bands for both price series finds similar (28%
to 35%) average percentage width differences at all 16 certainty levels.
In short, the non-normal models produce more precise predictions for the corn and soybean prices to
be expected at any given year. Realistic confidence bands for the actual price occurrences, however, are as
important in applied research. Figures 1 and 2 also show the 95% confidence bands for the soybean price
occurrences under the normal and non-normal models. In the normal case (Figure 1), the bands are wider than
those constructed for the predictions, but they are still symmetric about the price predictions. This is due to the
addition of the normally distributed error term. Note that two of the price realizations exceed the upper bound
while none is located even close to the lower bound, versus the theoretically required number of 1.25.
Examination of the 80 to 95% confidence bands reveals that, from 83% to 95%, no observations are
left below the lower bounds, and only two are beneath the 80% band versus the theoretically required number
of five (Table 4). When all of the 16 confidence bands are considered, only five out of a theoretically required
total of 50 observations are found below the lower bounds, while 58 are above the upper bounds (Table 4).
The assumption of error term normality, which causes these confidence bands to be symmetric, appears
incompatible with the soybean price data. The confidence bands for the soybean price occurrences under the
non-normal model (Figure 2) are markedly non-symmetric about the predictions, reflecting the kurtosis and
right-skewness of the estimated error term distribution. The 95% band leaves one observation above and one
below its boundaries (1.25 required). The, 90, 85 and 80% bands leave three and two, five and three, and six
and four observations below and above their boundaries, respectively, vs. the theoretically required numbers of
2.5, 3.75 and five, respectively. When all of the 16 confidence bands are considered, 59 and 42 observations
are found below and above their lower bounds, respectively, versus the theoretically required number of 50
(Table 4). Compatibility with the data is improved by lower bounds that are consistently higher than their
normal counterparts, and thus closer to the low price occurrences, combined with upper bounds that become
relatively higher than the normal bounds at reduced α levels (Table 2 and Figures 1 vs. 2). The non-normal
confidence bands are also narrower than the normal bands, on average and up to the 93% confidence interval.
On average, the 16 non-normal confidence bands for the corn price occurrences also appear more
compatible with the observed data than the bands under the normal-error model, leaving 52 and 46
observations below and above their lower bounds, respectively, vs. 28 and 45, respectively. They are also
narrower than the normal bands on average and up to the 93% confidence interval (Table 4).
The fact that the non-normal models result in confidence bands that are more consistent with the
observed data is important given the many empirical applications of confidence intervals. It also provides for
an intuitive explanation of the more precise slope parameter and dependent variable predictions afforded by the
non-normal models. That is, the degree of uncertainty about the location of the true regression line (or hyper
plane in the multiple regression case) is reduced by an improved accounting of the phenomena causing the data
deviations form the line (or hyper plane), i.e. by the improved modeling of the error term distribution.
An Empirical Model of the West Texas Cotton Basis
Another issue of importance to agricultural economists is the behavior of the basis in a futures
market, and measuring the impact of market and policy factors on the level and variability of the basis
(Seamon and Kahl). For the purposes of this study, the monthly (January 1980 to December 2000) West
Texas cotton basis is defined as the difference between the West Texas cash price and the U.S. futures
cotton price for the September contract.
Several factors have been hypothesized to affect this basis, including the monthly Texas, U.S., and
foreign production (TXP, USP, and FP) and beginning stocks (TXBS, USP, and FBS); monthly foreign
mill use (FMU); the U.S. price of rayon (PR); the opportunity cost of storage (STRC), as measured by
Seamon and Kahl; transportation costs measured by the monthly U.S. railroad index for farm products
(RRI); seasonal effects measured by dummy variables representing the planting (SD=0 for March to July)
versus the harvesting and marketing seasons (SD=1 for August to February); and agricultural policy
effects measured by dummy variables for the pre 1985 farm bill years (PD=1 from 1980 to 1985 and zero
otherwise), the 1985 farm bill period (PD1=1 from 1986 to 1995 and zero otherwise), and the post 1996
farm bill era (PD2=1 from 1996 to 2001 and zero otherwise).
All of the variables discussed above are stationary according to the augmented Dickey-Fuller unit
root test, with the exception of the price of rayon. The first difference in the price of rayon (FDPR), which
is stationary, is therefore used instead of PR. A Durbin-Watson test based on the OLS residuals reveals
first-order positive autocorrelation. The autocorrelation function of the OLS residuals is steadily
decreasing, while the partial autocorrelations become statistically insignificant at lag five. When a tenthorder autoregressive model is estimated using Gauss 386i autoreg procedure, the autocorrelation parameter
estimates decrease in size and the fifth and higher-order coefficients are not statistically different from zero at the
10% level. Thus, it is concluded that a fourth-order autoregressive [AR(4)] error term specification is sufficient to
correct for autocorrelation.
The standard maximum likelihood procedures outlined above [equation (4)] are then used to estimate an
AR(4) model under the assumption of error term normality and heteroskedasticity [NHAR(4)], shifting the
2
2
2
2
variance of the error term (σ ) by σ SD, σ PD1, and σ PD2, according to the seasonal and policy dummies (Table 5).
As in previous econometric analysis of regional basis, most of the regression parameters are insignificant. With
the exception of SD, the relatively large standard errors of the slope parameter estimators make it impossible to
ascertain the direction and magnitude of the impact of the proposed explanatory variables on the basis, with
reasonable statistical certainty.
The NHAR(4) model is then extended to account for error term non-normality [NNHAR(4)]. In this
case, Ramirez and Shonkwiler’s proposed procedures are expanded by letting the first non-normality parameter
(Θ) shift by ΘSD, ΘPD1, and ΘPD2, according to the seasonal and policy dummies. Since Θ affects both Skew[U]
and Kurt[U] [equation (3)], this model specification assumes an error term distribution that can have a different
variance, skewness and kurtosis depending on the season and policy period. The slope parameter estimators under
this NNHAR(4) model exhibit substantially lower standard errors than their normal-error counterparts (Table 5),
empirically validating the previously discussed Monte Carlo simulation results again. Final [FNHAR(4) and
FNNHAR(4)] models are obtained by excluding the transportation cost (RRI) and 1996-2001 policy period
(PD2) variables from the regression equation, since they show the lowest levels of statistical significance in both
of the initial models and their corresponding parameter estimates bear incorrect signs.
All parameter estimates in the final models have the signs that would be expected from theory (Nelson).
The estimates for the standard errors of the slope parameter estimators are again all higher under the normal
model. Eight of the eleven standard error estimates are over 70% higher; and six are more than twice as high as in
the non-normal model. On average, the standard error estimates under the NHAR(4) model are 99.6% higher. As
a result, in the NNHAR(4) model, six regression parameters are statistically significant at the 1% level, and two
more at the 5% level, while only one regression parameter is statistically significant under the NHAR(4) model.
These remarkable estimation efficiency gains are related to the substantial 70.47 increase in the
maximum log-likelihood function value obtained by the modeling of non-normality. To put this number into
2
perspective, recall that an increase of just χ (4,0.01)÷2 = 6.64 is required for the likelihood ratio test to reject the null
hypothesis that all non-normality parameters are equal to zero and conclude that the NNHAR(4) model represents
a statistically significant improvement over the NHAR(4) model. Such a large log-likelihood value increase is an
unmistakable indication that the observed basis data (Figure 3) is much more likely to have been generated by the
non-normal than by the normal-error model. The D’Agostino-Pearson normality test applied to the OLS
residuals rejects the null hypothesis of error term normality at the 1% level, supporting this conclusion.
The confidence bands for the basis predictions and occurrences are generated using the same numerical
procedures described in the case of the corn and soybean price models. As before, few key summary statistics
about the bands are more revealing than their graphical presentation. As a result of the larger standard errors, the
80 to 99% confidence bands for the predictions from the normal model are, on average, 63% wider than those
from the non-normal model, i.e. the later produces substantially more precise predictions of the West Texas
cotton basis. As in the corn and soybean price models, the NNHAR(4) confidence bands for the actual basis
occurrences are more “efficient” and theoretically consistent than the bands from the normal model (Table 6).
They are more efficient in the sense that they are substantially narrower (7.306 vs. 9.643 basis points), on
average, and at all 20 α levels evaluated, although, as before, they approach the width of the normal bands the
lowest α of 0.01. At the same time, the 80 to 99% non-normal confidence bands are more theoretically consistent
since the numbers of observations found below and above their lower and upper bounds are much closer to the
theoretically expected numbers (Table 6). The more reliable non-normal confidence bands of the West Texas
cotton basis are clearly useful for applied research and decision making about futures contracting.
Another interesting and original byproduct of the NNHAR(4) model is the finding that, in addition to
affecting the mean and variance, policy and seasonal variables can also shift the skewness and kurtosis of the
error term distribution and, thus, of the conditional distribution of an economic time series. The estimated
conditional distributions of the West Texas cotton basis for the two seasons under the three policy periods in
the analysis are presented in Figures 4 and 5, assuming the average values for all other explanatory variables
during the corresponding season and policy period.
Note the substantial kurtosis and right skewness of the distributions during the March to July planting
season. In the current, post-1985 Farm Bill period, for example, the distribution shows a mean of –2.62 and a
standard deviation of 2.83. It implies a 0.3% (5%) probability of a basis realization that is less (more) than
one standard deviation from the mean, versus the 16% expected under a normal distribution. The conditional
distributions of the West Texas cotton basis before and during the 1985 Farm Bill period had estimated means
of –5.77 and –1.46 points, and standard deviations of 3.55 and 3.07, respectively. They were somewhat less
kurtotic and asymmetric, with implied probabilities of basis realizations below (above) one standard deviation
from their means of 5.9% and 7.1% (9.1% and 10.3%), respectively. In short, the probability of a high
positive basis during the planting season was much higher during the 1985 Farm Bill period, and it is almost
negligible in the current policy period, as it could be inferred from the data (Figure 3).
The estimated conditional distributions of the West Texas cotton basis during the harvesting and
marketing season (Figure 5) are nearly normal under all the three policy periods analyzed. They show lower
means (–6.08 vs. –5.77, –2.43 vs. –1.46, and –3.74 vs. –2.62, respectively) and standard deviations (2.64 vs.
3.55, 2.13 vs. 3.07, and 1.59 vs. 2.83, respectively) than during the planting season. Because of the stronger
non-normality in the August-to-February distributions, however, these reduced standard deviations do not
imply a lower overall degree of dispersion in the distribution of the basis in this case (Figures 4 and 5).
This is empirically important since, under error term non-normality, normal-error models such as the
NHAR(4), provide consistent estimates for the error term variance (note the similarity in the variance
estimates from the FNHAR(4) and the FNNHAR(4) models in Table 5), which would be misinterpreted as
appropriate measures of the degree of dispersion of the distribution. For example, from the relatively large
2
negative value of the variance shifter for the harvesting and marketing season (σ SD) in the FNHAR(4) model,
one would falsely conclude that the distribution of the basis during this season exhibits a much lower overall
degree of dispersion about its mean value.
Concluding Remarks
Agricultural economics research often involves the estimation of regression models with a limited
amount of data, and more precise and realistic statistical inferences from these models are always useful.
As illustrated above, the non-normal error multiple regression model evaluated in this study can provide for
substantially more precise and realistic statistical inferences than the currently available estimation
techniques that assume normality. Since many dependent variables of interest to agricultural economists,
such as commodity prices, crop acreage, yields, product supply, profits, etc., are likely non-normally
distributed, this could become an important tool in agricultural economics research. The theory and
examples above also suggest that the degree of improvement over the normal-error models depends on how
much the true error term distribution deviates from normality and on how well the non-normal error term
distribution on which the partially adaptive estimation procedure is based approximates the true datagenerating distribution. This is why very flexible distributions, such as the expanded form of the Johnson
SU family, should be preferred for partially adaptive estimation. The applications above also illustrate that
proposed technique could be useful if the estimated models will be used to simulate conditional probability
distributions for the dependent variable, which are often used for economic risk analysis.
References
Bickel, P.J. (1976). Another look at robustness. Scandinavian Journal of Statistics, 3:145-168.
Bierens, H.J. (1981). Robust Methods and Asymptotic Theory in Nonlinear Econometrics, Springer
Verlag, Berlin.
D’Agostino, R.B., A. Belanger, and R.B. D’Agostino Jr. A Suggestion for Using Powerful and Informative
Tests of Normality. The American Statistician, 44(4, 1990):316-321.
Dielmam, T., and Pfaffenberger (1982). LAV (least absolute value) estimation in linear regression: a
review. TIMS Studies in the Management Sciences, 19:31-52.
Goldfeld, S.M., and R.E. Quandt. Econometric Modeling with Non-Normal Disturbances. Journal of
Econometrics, 17(1981):141-155.
Hsieh, D.A., and C.F. Manski. Monte Carlo evidence on adaptive maximum likelihood estimation of a
regression. Annals of Statistics, 15(1987):541-551.
Johnson, N.L., S. Kotz, and N. Balakrishnan. 1994. Continuous Univariate Distributions. New York:
Wiley & Sons.
Joiner, B.L., and D.L. Hall (1983). The ubiquitous role of f'/f in efficient estimation of location. The
American Statistician, 37:128-133.
Judge, G.G., W.E. Griffiths, R. Carter Hill, H. Lutkepohl, and Tsoung-Chao Lee. The Theory and
Practice of Econometrics, John Wiley & Sons, Inc., 1985.
-TKPUM[ + CPF #. 4QDD 1P #RRTQZKOCVKPI VJG 5VCVKUVKECN 2TQRGTVKGU QH 'NCUVKEKVKGU
4GXKGY QH
'EQPQOKEU CPF 5VCVKUVKEU McDonald, J.B., and S.B. White. A Comparison of some Robust, Adaptive, and Partially Adaptive
Estimators of Regression Models. Econometric Reviews, 12(1, 1993):103-124.
McDonald, J.B., and W.K. Newey. Partially Adaptive Estimation of Regression Models Via the
Generalized t Distribution. Econometric Theory, 4(1988):428-457.
Moss, C.B., and J.S. Shonkwiler. Estimating Yield Distributions Using a Stochastic Trend Model and NonNormal Errors. American J. Agricultural Economics, 75(November 1993): 1056-62.
Mosteller, F., and J.W. Tukey (1977). Data Analysis and Regression, Addison-Wesley, Reading,
Massachusetts.
Nelson, J. An Empirical Investigation of the Cotton Basis for the Southern High Plains of Texas. MS
Thesis, Dept. of Agricultural and Applied Economics, Texas Tech University, 2001, p. 36-40.
Newey, W.K. (1988). Adaptive estimation of regression models via moment restrictions. Journal of
Econometrics, 38:301-339.
Ramirez, O.A. Estimation and Use of a Multivariate Parametric Model for Simulating Heteroskedastic,
Non-Normal Random Variables: The Case of Corn-Belt Corn, Soybeans and Wheat Yields.
American J. Agricultural Economics 79(February 1997):191-205.
Seamon, V.F. and K.H. Kahl. An Analysis of Factors Affecting the Regional Cotton Basis. Paper
presented at the NCR-134 Conference on Applied Commodity Price Analysis, Forecasting, and
Market Risk Management; Chicago, Illinois, April 17-18, 2000.
Zeckhauser, R., and M. Thompson. Linear Regression with Non-Normal Error Terms. Review of
Economics and Statistics, 52(1970):280-286.
Table 1. Root Mean Squared Error (RMSE) of slope estimators (sample size = 50; number of samples =
500). Source: McDonald and White (1993).
Underlying Error-Term
Estimation
Normal
Technique
Underlying Error-Term
Normal
Log-
Estimation
Mixture
Normal
Technique
Normal
Normal
Log-
Mixture
Normal
OLS
0.28
0.28
0.28
Huber 1, c=1
0.29
0.14
0.16
LAD
0.35
0.13
0.17
Huber 1, c=1.5
0.28
0.16
0.18
BT
0.29
0.14
0.18
Huber 1, c=2
0.28
0.19
0.20
GT
0.30
0.12
0.12
Huber 2 c=1
0.56
0.12
0.12
T
0.28
0.11
0.12
Huber 2 c=1.5
0.41
0.11
0.15
BT, p≥1
0.29
0.13
0.17
Huber 2 c=2
0.32
0.13
0.16
GT, p≥1
0.30
0.12
0.12
Manski (AML)
0.28
0.12
0.13
EGB2(p=q)
0.28
0.12
0.15
Newey (j)
0.30
0.12
0.11
EGB2
0.29
0.12
0.05
Proposed
0.28
0.11
0.05
Notes: OLS is the Ordinary Least Squares estimator; LAD is the Least Absolute Deviations estimator
(Gentle, 1997); BT is the power exponential or Box-Tiao estimator (Zeckhauser and Thompson, 1970), GT
is a partially adaptive estimator based on the generalized t distribution (McDonald and Newey, 1984,
1988); t is a partially adaptive estimator based on the Student’s t distribution; EGB2 is a partially adaptive
estimator based on the exponential generalized beta distribution of the second kind; Huber 1 and 2 refer to
the estimators proposed by Huber (1964) and Huber (1981); Manski (AML) is the adaptive maximum
likelihood estimator advanced by Hsieh and Manski (1987) based on a normal kernel density; and Newey
(j) is the generalized method of moments estimator with j moments used in estimation (Newey, 1988). For
more details about the former estimation techniques please see McDonald and White (1993).
Table 2: RMSE of the 1000-sample slope-parameter estimates under eigth underlying error-term
distributions and the normal and the proposed partially adaptive estimator alternatives.
Underlying
Distribution
i.i.d. Var. Cont.
Normal
i.i.d. LogNormal
Het. Var. Cont.
Normal
Het. LogNormal
0.5 Aut. Var.
Cont. Normal
0.8 Aut. Var.
Cont. Normal
0.5 Aut. LogNormal
0.8 Aut. LogNormal
Normal Estimators
EGB2
i.i.d.
Heter.
Autoc.
i.i.d.
i.i.d.
Heter.
Autoc.
0.284
--
--
0.126
0.115
--
--
0.291
--
--
0.054
0.050
--
--
0.293
0.212
--
0.125
0.112
0.107
--
0.281
0.202
--
0.117
0.104
0.052
--
0.320
--
0.260
0.149
0.141
--
0.114
0.388
--
0.208
0.195
0.188
--
0.098
0.334
--
0.238
0.143
0.126
--
0.053
0.439
--
0.219
0.299
0.260
--
0.051
Proposed Estimator
Table 3. Parameter estimates, standard error estimates, and statistical significance of parameters of normal
and non-normal error regression models for corn and soybean prices.
OLS
AR(1)
SUR-AR(1)
NSUR-AR(1)
NNSUR-AR(1)
Par. Est. S.E. Est. Par. Est. S.E. Est. Par. Est. S.E. Est. Par. Est. S.E. Est. Par. Est. S.E. Est.
θC
--
--
--
--
--
--
--
--
µC
BC0
--
--
--
--
--
--
--
--
3.0840 0.1423** 3.0860
0.2263** 3.0817**
0.2273
BC1
-1.5611 1.3432ns -1.4552
2.1317ns -1.4237ns
2.1879 -2.1217 0.5563** -2.3509 0.3929**
BC2
-1.0001 2.6509ns -1.3787
4.1937ns -1.4244ns
4.3186
σC
0.3490
--
0.2811
ρC
--
--
0.5460
θS
--
--
--
--
--
--
--
--
0.5677 0.1600**
µS
BS0
--
--
--
--
--
--
--
--
15.7161 5.0376**
5.2544 0.3246** 5.3045
0.4998**
BS1
11.8443 3.0636** 11.5373
4.7101**
11.6223 4.0533** 11.1429 3.8273** 10.1967 2.8094**
BS2
-25.4146 6.0462** -25.0758 9.2701**
-25.1588 7.9850** -24.1720 7.4968** -21.4259 5.3032**
--
3.1351 0.1600**
--
0.9446 0.2875**
--
--
3.1722 0.1065**
--
--
--
0.2818**
0.0282
0.2821 0.0282**
0.2990 0.0804**
0.1185** 0.5347**
0.1111
0.5353 0.1112**
0.5688 0.0987**
5.3262 0.4172**
5.2899 0.4291**
5.3607 0.3347**
σS
0.7960
--
0.6587
--
0.6645 0.0677**
0.6645 0.0677**
0.7369 0.1603**
ρS
--
--
0.5132
0.1214**
0.4066 0.1227**
0.4073 0.1232**
0.4484 0.0763**
ρCS
--
--
-MVCLF
-33.86
0.3598 0.1310** 0.3597 0.1311** 0.4644 0.1158**
MVCLF 36.98 MVCLF 36.92 MVCLF 52.54
R C=0.44 R S=0.28 R C=0.61 R S=0.48
R C=0.61 R S=0.47 R C=0.61 R S=0.47 R C=0.61R S=0.47
2
R
2
2
2
2
2
2
2
2
2
2
Notes: MVCLF stands for the maximum value reached by the concentrated log-likelihood function. Par.
Est. and S.E. Est. refer to the parameter and standard error estimates, respectively. The parameter and
standard error estimates corresponding to BC1 and BS1, and to BC2 and BS2 have been divided by 100 and
10000, respectively. * and ** denote statistical significance and the 90 and 95% level, respectively,
2
according to two-tailed t tests. The R ’s are calculated by dividing the regression sums of squares (based on
the autocorrelated {AR(1)} predictions) by the total sums of squares, i.e. it are the square of the correlation
coefficients between the AR(1) predictions and the observed corn (c) and soybean (s) prices.
Table 4*. Select statistics about the 80 to 95% confidence bands for the price occurrences under the normal and
non-normal corn and soybean price models.
%CI
80%
81%
82%
83%
84%
85%
86%
87%
88%
89%
90%
91%
92%
93%
94%
95%
Sum/Ave
TR#
5.00
4.75
4.50
4.25
4.00
3.75
3.50
3.25
3.00
2.75
2.50
2.25
2.00
1.75
1.50
1.25
50
%CI
80%
81%
82%
83%
84%
85%
86%
87%
88%
89%
90%
91%
92%
93%
94%
95%
Sum/Ave
#TR
5.00
4.75
4.50
4.25
4.00
3.75
3.50
3.25
3.00
2.75
2.50
2.25
2.00
1.75
1.50
1.25
50
Normal Corn Price Model
#ULB #OUB ALB AUB AWCB
4
3
2.249 2.993 0.744
3
3
2.240 3.001 0.761
3
3
2.231 3.010 0.779
2
3
2.222 3.019 0.797
2
3
2.213 3.029 0.817
2
3
2.202 3.039 0.837
2
3
2.191 3.050 0.859
2
3
2.180 3.061 0.881
1
3
2.168 3.074 0.905
1
3
2.155 3.087 0.931
1
3
2.141 3.100 0.959
1
3
2.126 3.116 0.989
1
3
2.109 3.132 1.023
1
2
2.091 3.151 1.060
1
2
2.070 3.171 1.101
1
2
2.047 3.195 1.149
28
45
2.165 3.077 0.912
Normal Soybean Price Model
#ULB #OUB ALB AUB AWCB
2
5
5.220 6.989 1.768
2
5
5.200 7.009 1.809
1
5
5.179 7.030 1.851
0
5
5.157 7.052 1.895
0
5
5.134 7.075 1.941
0
5
5.109 7.099 1.990
0
4
5.083 7.125 2.041
0
4
5.057 7.152 2.095
0
3
5.028 7.181 2.153
0
3
4.997 7.212 2.214
0
3
4.964 7.245 2.280
0
3
4.929 7.280 2.352
0
2
4.889 7.320 2.432
0
2
4.844 7.364 2.520
0
2
4.795 7.413 2.618
0
2
4.738 7.470 2.732
5
58
5.020 7.189 2.168
Non-Normal Corn Price Model
#ULB #OUB ALB AUB AWCB
7
3
2.315 2.910 0.595
7
3
2.305 2.920 0.615
7
3
2.294 2.931 0.636
7
3
2.283 2.942 0.659
5
3
2.270 2.955 0.684
4
3
2.257 2.968 0.711
4
3
2.243 2.982 0.740
2
3
2.227 2.998 0.771
2
3
2.210 3.015 0.806
2
3
2.191 3.035 0.844
1
3
2.169 3.056 0.887
1
3
2.145 3.080 0.935
1
3
2.117 3.108 0.991
1
3
2.085 3.141 1.056
1
2
2.046 3.180 1.134
0
2
1.998 3.228 1.230
52
46
2.197 3.028 0.831
Non-Normal Soybean Price Model
#ULB #OUB ALB AUB AWCB
6
4
5.410 7.001 1.592
5
4
5.401 7.035 1.634
5
4
5.391 7.071 1.680
5
4
5.382 7.110 1.728
5
4
5.372 7.151 1.779
5
3
5.362 7.195 1.833
5
3
5.351 7.243 1.891
5
2
5.340 7.295 1.955
4
2
5.328 7.352 2.024
3
2
5.316 7.414 2.098
3
2
5.303 7.483 2.180
3
2
5.289 7.561 2.272
2
2
5.273 7.650 2.377
1
2
5.256 7.752 2.496
1
1
5.237 7.875 2.637
1
1
5.216 8.021 2.805
59
42
5.327 7.388 2.061
*See notes below Table 5 for the definitions of the labels and summary statistics presented in Table 2.
Table 5. Parameter estimates, standard error estimates, and statistical significance of parameters of normal and
non-normal error regression models for the West Texas cotton basis.
Par.
Var.
-ρ1
-ρ2
-ρ3
-ρ4
B0
-B1
TXP
B2
USP
B3
FP
B4
TXBS
B5
USBS
B6
FBS
B7
FMU
B8
RRI
B9
STRC
B10
FDPR
B11
SD
B12
PD1
B13
PD2
2
-σ
2
SD
σ SD
2
σ PD1 PD1
2
σ PD2 PD2
-Θ
-µ
SD
ΘSD
PD1
ΘPD1
PD2
ΘPD2
MVLLF
2
R
NHAR(4)
P.E.
S.E.
1
0.9807
0.0687
2
-0.2385
0.0965
0.1319
0.0975
3
-0.1073
0.0612
-14.5204 12.2673
0.1930
0.5159
0.3022
0.2476
0.0913
0.1452
-0.3323
0.3043
0.1800
0.3508
0.0162
0.1053
-0.0361
0.3599
0.0146
0.1479
-1.4566
1.2934
1.0527
0.6739
1
-1.0991
0.4225
2.1954
2.3614
1.1976
2.6599
1
16.0036
2.8967
1
-11.3179
2.8122
-0.3919
1.3692
1
-3.4360
1.0000
-----------352.820
0.77
NNHAR(4)
P.E.
S.E.
1
0.96261 0.02720
1
-0.13640 0.04190
1
0.13418 0.04518
1
-0.15764 0.03732
-13.7478 8.60659
-0.11620 0.13143
2
0.31761 0.14303
1
0.23745 0.08869
3
-0.47715 0.25815
1
0.52388 0.16576
2
0.16266 0.06824
2
-0.26306 0.11320
0.03185 0.08302
-1.15475 1.26164
0.43163 0.47683
1
-0.69499 0.23655
1.85166 1.84490
-0.85989 2.30983
1
11.35527 4.13477
-5.82120 3.87761
-1.76188 1.42804
1
-3.91090 1.35165
1
0.95789 0.22514
1
0.69869 0.25017
1
-1.21066 0.30811
0.02153 0.19116
1
0.57618 0.19320
-282.353
0.77
FNHAR(4)
P.E.
S.E.
1
0.98904 0.06593
1
-0.24666 0.09358
0.13667 0.09606
3
-0.11249 0.06017
2
-18.26522 7.84887
0.18625 0.46566
0.33011 0.23807
0.10201 0.13689
-0.32229 0.29186
0.18375 0.34317
0.03877 0.10515
0.02261 0.17314
---1.35748 1.25214
1.03744 0.67141
2
-1.06572 0.41410
1.29106 1.15962
--1
16.13068 3.00359
1
-11.50518 2.77815
-0.38268 1.54083
1
-3.35245 1.02157
-----------352.964
0.77
FNNHAR(4)
P.E.
S.E.
1
0.95853 0.02773
1
-0.13685 0.04244
1
0.13438 0.04602
1
-0.15605 0.03682
3
-11.83432 6.66378
-0.12095 0.12967
1
0.34039 0.08453
1
0.24439 0.06782
2
-0.48446 0.24536
1
0.55098 0.15591
1
0.17332 0.04436
1
-0.26656 0.10223
-1.13765
0.35747
-0.63615
2.38576
1.23339
0.38805
1
0.18115
2
1.11527
10.95924
-5.42047
-1.80659
-3.89927
0.93976
0.72557
-1.19269
3.63400
3
3.34320
1.40307
1
1.33951
1
0.20077
1
0.26177
1
0.29155
0.57207
-282.448
0.77
1
1
0.17190
Notes: NHAR(4), NNHAR(4), FNHAR(4) and FNNHAR(4) refer to the initial normal and non-normal and to the
final normal and non-normal heteroskedastic fourth-order autoregressive models; the parameters (Par.) and
1 2
variables (Var.) are as defined in the text; P.E. and S.E. refers to the parameter and standard error estimates; , ,
and
3
denote statistical significance at the 1%, 5% and 10% level, respectively, according to two-tailed t-tests;
2
MVLLF refers to the maximum value reached by the model’s concentrated log-likelihood function; and the R is
calculated by dividing the regression sum of squares (based on the autocorrelated {AR(4)} predictions) by the
total sum of squares, i.e. it is the square of the correlation coefficient between the AR(4) predictions and the
observed basis values.
Table 6. Select statistics about the 80 to 99% confidence bands for the basis occurrences under the final normal
{FNHAR(4)} and non-normal {FNNHAR(4)} West Texas cotton basis models.
%CI
80%
81%
82%
83%
84%
85%
86%
87%
88%
89%
90%
91%
92%
93%
94%
95%
96%
97%
98%
99%
Sum/
Ave
TR#
25.20
23.94
22.68
21.42
20.16
18.90
17.64
16.38
15.12
13.86
12.60
11.34
10.08
8.82
7.56
6.30
5.04
3.78
2.52
1.26
#ULB
11
11
10
9
9
9
9
9
7
7
5
3
3
3
3
3
3
2
2
1
265
119
FNHAR(4) MODEL
#OUB ALB AUB AWCB
14
-6.483 0.697 7.180
14
-6.566 0.779 7.345
13
-6.652 0.865 7.516
11
-6.741 0.954 7.695
11
-6.835 1.048 7.883
11
-6.934 1.146 8.081
11
-7.038 1.250 8.289
11
-7.149 1.361 8.509
8
-7.267 1.477 8.744
8
-7.392 1.601 8.994
7
-7.525 1.736 9.261
7
-7.672 1.882 9.554
7
-7.833 2.041 9.874
7
-8.012 2.219 10.230
5
-8.213 2.418 10.631
5
-8.447 2.649 11.096
4
-8.722 2.923 11.644
4
-9.065 3.263 12.328
1
-9.531 3.724 13.255
1
-10.28 4.473 14.755
160
-7.718
1.925
9.643
#ULB
21
19
19
19
18
17
17
16
16
16
15
14
11
7
5
5
5
3
2
1
246
FNNHAR(4) MODEL
#OUB ALB
AUB
27
-5.241 -0.456
24
-5.290 -0.370
22
-5.342 -0.278
21
-5.396 -0.181
20
-5.453 -0.076
20
-5.512 0.035
18
-5.577 0.156
15
-5.645 0.287
15
-5.719 0.431
14
-5.799 0.589
12
-5.886 0.764
12
-5.981 0.960
10
-6.089 1.185
9
-6.210 1.446
8
-6.350 1.753
7
-6.518 2.132
4
-6.725 2.613
4
-6.997 3.274
2
-7.399 4.288
1
-8.143 6.291
265
-6.064
1.242
AWCB
4.785
4.920
5.064
5.215
5.377
5.547
5.733
5.932
6.150
6.389
6.650
6.941
7.274
7.656
8.104
8.649
9.338
10.271
11.687
14.434
7.306
Notes: #TR refers to the number of observations that would be theoretically required to be below and above the
boundaries of the confidence band; #ULB and #OUB are the actual numbers found under and over the lower and
upper bounds, respectively; ALB, AUB, and AWCB stand for the average of the n=50 lower and upper bounds
and widths of the 50 confidence intervals comprising each of the 16 bands, respectively; and Sum/Ave. refer to the
sums (for #TR, #ULB, and #OUB) or averages (for ABL, AUB, and AWCB) across the 16 confidence bands.
Figure 1: 95% Confidence Bands for the Soybean Price Predictions and
Occurrences vs. Data under the Normal Regression Model
Soybean Price ($/bushel)
10
Bands for the Occurrences
9
Bands for the Predictions
8
Predictions
7
6
5
98
19
95
19
92
19
89
19
86
19
83
19
80
19
77
19
74
19
71
19
68
19
65
19
62
19
59
19
56
19
53
19
19
50
4
Year
Figure 2: 95% Confidence Bands for the Soybean Price Predictions and
Occurrences vs. Data under the Non-Normal Regression Model
Soybean Price ($/bushel)
10
Bands for the Occurrences
9
8
Bands for the Predictions
Predictions
7
6
5
Year
98
19
95
19
92
19
89
19
86
19
83
19
80
19
77
19
74
19
71
19
68
19
65
19
62
19
59
19
56
19
53
19
19
50
4
Figure 3: Monthly 1980-2001 West Texas Cotton Basis Data
30
25
Basis (Points)
20
15
10
5
0
-5
-10
-15
Year
Post 1985 Farm Bill Period
Pre 1985 Farm Bill Period
Basis (points)
4
3
2
1
0
-1
-2
-3
-4
-5
-6
-7
-8
-9
0
-1
1
1985 Farm Bill Period
-1
2
0.20
0.18
0.16
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0.00
-1
Probability
Figure 4: Estimated Probability Distributions for the West Texas
March-to-July Cotton Basis for three Policy Periods
2001
2000
1999
1998
1997
1996
1995
1994
1993
1992
1991
1990
1989
1988
1987
1986
1985
1984
1983
1982
1981
1980
-20
Post 1985 Farm Bill Period
Basis (points)
4
3
2
1
0
-1
-2
-3
-4
-5
-6
1985 Farm Bill Period
-7
-8
-9
0
-1
1
Pre 1985 Farm Bill Period
-1
2
0.20
0.18
0.16
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0.00
-1
Probability
Figure 5: Estimated Probability Distributions for the West Texas
August-to-February Cotton Basis for three Policy Periods